Kayak does not sell anything.....
Best option is to find an airline that might sell that ticket (which I doubt that you will find one...
If looking for a better deal try searching for DEL->SYD->LAX or BKK-->SYD->LAX or SIN->SYD->LAX (or even China/ HK/Dubai stops) with +- 3 days option (first page of Kayak). Also, you will have to buy KTM to your first destination ticket on your own. (Don’t forget to include that on your price).
If not then try and find an agency and you will be fine. Personally I always do that as it’s cheap(er). However, I have seen a few people complaining about them. I think your bad luck needs to be really bad for you to get into that kind of situation, and it seems that’s not the case with you as you are about to travel the world. So….Good luck...
,Also make sure that you have the credit card that you are buying the ticket with in hand all the time. Anyone can ask that at the time of issuing a boarding pass no matter how you buy.